Of late, there has been a dramatic growth in the capacity of wireless communication networks—Cellular networks have grown from analog “voice-only” systems to current 3rd Generation networks that provide a maximum download capacity of 2 Mbps—catering to voice, data and multimedia services; Wireless LANs have evolved from initial data rates of 2 Mbps specified by the IEEE 802.11-99 specification [refer to “Local and Metropolitan Area Networks—Specific Requirements—Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications”, IEEE Std802.11-1999, IEEE, August 1999] to the present IEEE 802.11a specification [refer to “Local and Metropolitan Area Networks—Specific Requirements—Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications: Higher-Speed Physical Layer Extension in the 5 GHz Band”, IEEE Std 802.11a-1999, IEEE, September 1999] that provide link rates of up to 54 Mbps.
To satiate the need for greater data rates, standardization for a are currently developing the next generation wireless standards.
<MIMO>
At the core of enhancing the capacity of several next generation wireless systems is MIMO—multiple-input-multiple-output—a technology that when applied to wireless communications employs the use of NT transmit antennas and NR receive antennas to better effect communication. The use of multiple antennas offers the flexibility of choosing from spatial-multiplexing gain—where a dramatic increase in spectral efficiency, up to min(NT, NR) times that of a conventional single antenna (SISO) system [refer to “On limits of wireless communications in fading environments when using multiple antennas,” Wireless Personal Communications, pp. 36-54, March 1998] can be realized; or, diversity gain—where up to NTNR paths that exist between transmitter and receiver may be used to exploit the diversity in the channel, leading to higher link-reliability in the wireless channel. In general, there are tradeoffs between increased data-rate (spatial-multiplexing) and increased reliability (diversity).
<Feedback>
In realistic scenarios, wireless communication systems suffer from a range of impairments. These range from non-ideal device behavior in the transceiver itself, to variability/selectivity of the channel—in the time, frequency and spatial domains. Feedback in a communications system can enable the transmitter to exploit channel conditions and avoid interference. In the case of a MIMO channel, feedback can be used to specify a pre-coding matrix at the transmitter that facilitates the exploitation of the strongest channel modes, or, the inherent diversity of the channel [refer to “What is the Value of Limited Feedback for MIMO Channels?”, IEEE Communications Magazine pp. 54-59, October 2004].
An example of a closed loop MIMO system is one implementing eigen-mode spatial-multiplexing—where the transmitter and receiver, having channel state information (CSI), use a transform such as the singular value decomposition (SVD), to convert the MIMO channel into a bank of scalar channels, with no cross-talk between channels [refer to “Promises of Wireless MIMO Systems,” http://www.signal.uu.se/courses/semviewgraphs/mw—011 107.ppt]. Eigen-mode spatial-multiplexing is an optimal space-time processing scheme in the sense that it achieves full diversity and full multiplexing gains of the channel [refer to “Transmitter Strategies for Closed-Loop MIMO-OFDM,” PhD thesis submitted to School of Electronic and Computer Engineering, Georgia Institute of Technology, July 2004]. The detection complexity of the eigen-mode spatial-multiplexing scheme increases only linearly with the number of antennas (in contrast to the optimum maximum likelihood sequence estimator (MLSE), which is open-loop but has an exponential complexity, rendering it intractable for implementation in practical systems).
In order to realize the benefits of eigen-mode spatial-multiplexing a.k.a. eigen-beamforming, CSI is required at the transmitter. An intuitive way of achieving this is to merely feedback the estimated channel state to the transmitter. However, feedback detracts from the payload carrying capacity of the system and is hence an expense, which must be minimized.
In the above-mentioned PhD thesis, the application of eigen-beamforming to a time division duplex (TDD) system is described. Although the ‘over the air channel’ is reciprocal, the cascading of different transmit and receive RF chains on both ends of the link render the base-band channel non-reciprocal. Once appropriate calibration (not described in the thesis) is performed, the transmit and receive-filter matrices required to facilitate eigen-beamforming are described by simple reciprocal arrangements at both ends of the link.
In US Patent Application Publication 2004/0082356A1, the authors describe a calibration scheme and the use of eigen-beamforming in the context of a WLAN network. In the described system, calibration is performed through the explicit feedback of channel estimates derived by the terminal from a training sequence transmitted by the base-station. The base-station obtains a similar estimate of the reverse channel from a training sequence transmitted by the terminal and computes a set of calibration coefficients that are explicitly fed back to the terminal and used by both the base-station and the terminal to render the base-band channel reciprocal. Once calibrated, channel decomposition (based on channel estimates derived from the training sequence transmitted by the base-station) is performed by the terminal to derive a set of transmit and receive filters required for eigen-beamforming by the terminal. Feedback of these filters to the base-station is performed implicitly, by means of a specially modulated training sequence, known in “System Description and Operating Principles for High Throughput Enhancements to 802.11,” doc: IEEE 802.11-04/870r0, as a steered sequence, from which the base-station can directly derive its receive-filter and correspondingly (from the reciprocality principle), its transmit-filter.
As TDD renders smaller capacities owing to the need for large guard-bands to counter the channel delay-spread, particularly in macro-cellular environments, cellular systems are prevalently frequency division duplex (FDD) [refer to “Comprehending the technology behind the UMTS wideband CDMA physical layer,” RF Signal Processing pp. 50-58, November 2002]. Although the uplink and downlink channels in FDD cellular systems are correlated to the extent that they typically share similar delay-spreads and power-delay profiles, for all other practical intents and purposes, they are considered uncorrelated. Hence, in order to perform eigen-beamforming, FDD systems cannot make use of feedback schemes that exploit channel reciprocality. Explicit feedback of the channel coefficients or related information must be used instead.
In US Patent Application Publication 2004/0234004A1, the authors describe a transceiver scheme whereby channel decomposition is performed in the frequency-domain, but transmit and receive filtering associated with eigen-mode spatial-multiplexing is performed in the time domain. The described receiver estimates the channel coefficients and performs a singular-value decomposition in order to derive a set of receive steering vectors. The channel coefficients are fed-back to the transmitter, which performs a second singular value decomposition in order to derive the appropriate set of transmit steering vectors. In general, the described system explicitly calls for the feedback of the channel coefficients from the receiver to the transmitter.
In US Patent Application Publication 2003/0235255A1, the authors describe methods by which water-filling may be used to enhance the capacity of an eigen-mode spatially multiplexed system. In order to realize the channel eigen-modes and corresponding transmit and receive filters, the specification describes a process similar to the above-mentioned US Patent Application Publication 2004/0234004A1, whereby the channel coefficients are explicitly fed back from transmitter to receiver.
In US Patent Application Publication 2004/0203473A1, the author describes methods by which a receiver may compute a bounded set of eigen-vectors, facilitating quantization over limited ranges of number-space for a system with two transmit antennae. While the proposed method achieves a reduction in feedback information by selecting a solution of eigen-vectors that results in relationships between elements of individual eigen-vectors, the method is limited to a system with two transmit antennae.
<Singular Value Decomposition>
The objective of eigen-mode spatial-multiplexing is to diagonalize the channel, rendering a vector channel into a group of individual scalar channels, there being no cross-talk between spatial channels (eigen-modes). The optimum transmit and receive steering matrices can be found using a singular value decomposition, as described in the following.
Assuming an NRx×NTx matrix [H]: [H] can be represented as a product of matrices of the form specified in equation (1) [refer to “Singular Value Decomposition,” http://mathworld.wolfram.com/SingularValueDecomposit ion.html],[H]=[U]·[D]·[V]H  (1)
where, [U] and [V] are unitary matrices of the left and right-handed singular-vectors and having dimension NRX×NRX and NTX×NTX, respectively; and [D] is an NRX×NTX matrix containing the singular-values of [H] along its diagonal. It may be worthwhile to note that there are min(NTX,NRX) positive non-zero singular-values of [H], the remainder of the elements of [D] being zero. Each positive singular-value corresponds to the gain on the corresponding spatial-mode (or eigen-beam) of the channel.
In the context of equation (1) and the remainder of this specification, the notation [A]H for a matrix [A] denotes the Hermitian of the matrix [A].
The singular-values and singular-vectors of a matrix are closely related to its eigen-values. In the context of the matrix [H] in equation (1), [U] and [D] correspond to the matrices of eigen-vectors and positive square roots of the eigen-values, respectively, of the left-handed matrix product [H]·[H]H; while [V] and [D] correspond to the matrices of eigen-vectors and positive square roots of the eigen-values of the right-handed matrix product [H]H·[H].
Based on the definitions above, it is further worthwhile to note that in the context of a non-square matrix [H], the singular-vector matrices [U] and [V] contain some trivial singular-vectors, corresponding to singular-values of zero. As such, there exists an ‘economy-size’ singular value decomposition in which [D] is always a square diagonal matrix of dimension corresponding to min(NTX,NRX) and [U] and [V] are matrices of the non-trivial singular-vectors of the system.
<Eigen-Mode Spatial-Multiplexing>
In a system based on eigen-mode spatial-multiplexing, the receiver estimates the channel from the transmitter to the receiver—[H], and performs a singular value decomposition to determine the matrix of left-handed singular-vectors—[U]. The receiver also feeds back the channel state information (e.g.: [H], as per US Patent Application Publication 2004/0234004A1 and US Patent Application Publication 2003/0235255A1, both of which are mentioned above) to the transmitter, which in turn performs a second singular value decomposition to determine the matrix of right-handed singular-vectors—[V].
Assuming that the transmitter performs spatial-multiplexing, transmitting data [x], the received signal, [y], can be represented by equation (2)[y]=[H]·[x]+[n]  (2)
where, [n] represents noise, which in the context of wireless systems, is typically modeled as an additive white Gaussian variable with finite power.
An open-loop receiver, for example the zero-forcing (ZF) detector, would determine an estimate of the transmitted data as:[{circumflex over (x)}]=[H]−1·[y]  (3)
The problem with such an approach is the noise-enhancement effect, which results in a signal-to-noise-ratio (SNR) degradation at the receiver [refer to “Digital Communications 3ed”, McGraw-Hill, March 1995].
In order to effect eigen-mode spatial-multiplexing, the transmitter pre-filters the data [x], with a transmit steering matrix—[V], and the receiver applies a matched-filter—[U]H to the received signal, [y]. Equation (4) represents the received signal, while equation (5) depicts the matched filtering applied by the receiver to estimate the transmitted data.[y]=[H]·[V]·[x]+[n]  (4)[{circumflex over (x)}]=[U]H·[y]  (5)
Expanding [H] as per equation (1), we obtain equation (6):[{circumflex over (x)}]=[D]·[x]+[U]H·[n]  (6)
It can be seen from equation (6) that the eigen-beamforming method results in perfect decoupling (i.e. no cross-talk) between streams and an SNR gain proportional to the square of the singular-values of the channel. The SVD method can be applied to any size and any rank of channel matrix since the SVD exists for any matrix [refer to “Singular Value Decomposition,” http://mathworld.wolfram.com/SingularValueDecomposit ion.html, which is mentioned above]. Finally and most importantly, eigen-mode spatial-multiplexing is optimal in the information theoretical sense since unitary filters preserve information.